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Let V be the vector space over the field of real numbers consisting of all real-valued sequences (a1, a2, a3, ...). Indicate, without proof, which of the following subsets of V is a vector space. (a) The set of all sequences satisfying ai = a2 = a3. (b) The set of all sequences that are increasing, i.e., for which aj < a2 < a3 < ... (c) The set of unbounded sequences. (d) The set of bounded sequences. (e) The set of sequences satisfying an+1 = an + an-1 for all n > 2. (f) The set of sequences satisfying an+1 = an × an-1 for all n > 2. Some comments: (i) You do not need to justify your answer, just write which of (a)-(f) are vector spaces and which are not. (ii) A sequence (an)n>1 is said to be bounded if there is a real number B > 0 (depending on the sequence) for which all a, lie in the interval [-B, B]. It is said to be unbounded otherwise.

Let V be the vector space over the field of real numbers consisting of all real-valued sequences (a1, a2, a3, ...). Indicate, without proof, which of the following subsets of V is a vector space. (a) The set of all sequences satisfying ai = a2 = a3. (b) The set of all sequences that are increasing, i.e., for which aj < a2 < a3 < ... (c) The set of unbounded sequences. (d) The set of bounded sequences. (e) The set of sequences satisfying an+1 = an + an-1 for all n > 2. (f) The set of sequences satisfying an+1 = an × an-1 for all n > 2. Some comments: (i) You do not need to justify your answer, just write which of (a)-(f) are vector spaces and which are not. (ii) A sequence (an)n>1 is said to be bounded if there is a real number B > 0 (depending on the sequence) for which all a, lie in the interval [-B, B]. It is said to be unbounded otherwise.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.1: Vector Spaces And Subspaces
Problem 7EQ
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Let V be the vector space over the field of real numbers consisting of all real-valued sequences (a1, a2, a3, ...). Indicate, without proof, which of the following subsets of V is a vector space. (a) The set of all sequences satisfying ai = a2 = a3. (b) The set of all sequences that are increasing, i.e., for which a1 < a2 < az < …. (c) The set of unbounded sequences. (d) The set of bounded sequences. (e) The set of sequences satisfying a,+1 = an + an-1 for all n > 2. (f) The set of sequences satisfying an+1 = an × an-1 for all n > 2. Some comments: (1) You do not need to justify your answer, just write which of (a)-(f) are vector spaces and which are not. (ii) A sequence (a,n)n21 is said to be bounded if there is a real number B > 0 (depending on the sequence) for which all a, lie in the interval [-B, B]. It is said to be unbounded otherwise.
Transcribed Image Text:Let V be the vector space over the field of real numbers consisting of all real-valued sequences (a1, a2, a3, ...). Indicate, without proof, which of the following subsets of V is a vector space. (a) The set of all sequences satisfying ai = a2 = a3. (b) The set of all sequences that are increasing, i.e., for which a1 < a2 < az < …. (c) The set of unbounded sequences. (d) The set of bounded sequences. (e) The set of sequences satisfying a,+1 = an + an-1 for all n > 2. (f) The set of sequences satisfying an+1 = an × an-1 for all n > 2. Some comments: (1) You do not need to justify your answer, just write which of (a)-(f) are vector spaces and which are not. (ii) A sequence (a,n)n21 is said to be bounded if there is a real number B > 0 (depending on the sequence) for which all a, lie in the interval [-B, B]. It is said to be unbounded otherwise.
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